Using Time-Frequency and Wavelet Analysis to Assess ...Using Time-Frequency and Wavelet Analysis to Assess Turbulence/Rotor Interactions Preprint November 1999 • NREL/CP-500-27151

Using Time-Frequency andWavelet Analysis to AssessTurbulence/Rotor Interactions

Preprint

November 1999 • NREL/CP-500-27151

N.D. Kelley and R.M. OsgoodNational Renewable Energy Laboratory

J.T. Bialasiewicz and A. JakubowskiUniversity of Colorado at Denver

To be presented at the 19th American Society ofMechanical Engineers (ASME) Wind EnergySymposiumReno, NevadaJanuary 10–13, 2000

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1

USING TIME-FREQUENCY AND WAVELET ANALYSIS*†

TO ASSESS TURBULENCE/ROTOR INTERACTIONS

Neil D. Kelley‡

Richard M. Osgood§

National Renewable Energy LaboratoryGolden, Colorado

Jan T. Bialasiewicz¶

Andrzej Jakubowski#

University of Colorado at DenverDenver, Colorado

ABSTRACTLarge loading events on wind turbine rotor

blades are often associated with transient bursts ofcoherent turbulent energy in the turbine inflow.These coherent turbulent structures are identified aspeaks in the three-dimensional, instantaneous,turbulent shearing stress field. Such organizedinflow structures and the accompanying rotoraeroelastic responses typically have time scales ofonly a few seconds and therefore do not lendthemselves for analysis by conventional Fourierspectral techniques. Time-frequency analysis (andwavelet analysis in particular) offers the ability tomore closely study the spectral decomposition ofshort period events such as the interaction of coherentturbulence with a moving rotor blade.

In this paper, we discuss our initial progress inthe application of time-frequency analysis techniquesto the decomposition and interpretation ofturbulence/rotor interaction. We discuss the resultsof applying both the continuous and discrete wavelettransforms for our application. Several examples aregiven of the techniques applied to both observedturbulence and turbine responses and those generatedusing numerical simulations. We found that thepresence of coherent turbulent structures, as revealedby the inflow Reynolds stress field, is a majorcontributor to large load excursions. These bursts ofcoherent turbulent energy induce a broadband

* This paper is a declared work of the U.S. Government and is not

subject to copyright protection in the United States.† This work has been supported by the U.S. Department of Energy

under contract no. DE-AC36-98-GO10337.‡ Principal Scientist, National Wind Technology Center, Senior

AIAA Member§ Senior Engineer, National Wind Technology Center¶ Professor, Department of Electrical Engineering# Graduate Student, Department of Electrical Engineering

aeroelastic response in the turbine rotor as it passesthrough them.

INTRODUCTIONIdentifying the processes responsible for

inducing large loading events on wind turbine rotorsremains one of the major unanswered questions ofthe technology. We know that such events can occurduring large yaw excursions or during periods ofsustained operation at large yaw angles. Previouslywe identified large loading events resulting from theinteraction of the turbine rotor with organized orcoherent turbulent structures in the inflow.1 Wefound that the number of cycles contained in thehigh-loading tails of spectral distributions scales withthe hub-height or local mean shearing stress u* andthe turbine layer vertical stability. The meanshearing stress or friction velocity is defined as

''* wuu =

where u' and w' are the longitudinal and vertical zero-mean wind components aligned with the mean shearvector. The turbine layer is defined as the verticaldistance between the ground and the maximumelevation of the rotor. The vertical stability isexpressed by the gradient Richardson number definedas

[ ]2m z)/(/)/()/( ∂∂∂∂= uzgRi θθ ,

where g is the gravity acceleration, z the height inmeters, θ m the layer mean thermodynamic potentialtemperature (K) given by

[ ] 28601000()()( ./p(z)zTz =θ ,

and T(z) and p(z) are the temperature (K) andbarometric pressure (hPa) at height z.

2

Figure 1. Example of rotor flapwise load excursions and corresponding hub-height, instantaneous Reynoldsstresses

The covariance, u'w' , represents one componentof the mean Reynolds stresses in turbulent flow. Theothers include u'v' and v'w' , where v' is zero-mean,lateral wind component. In a well-mixed,homogeneous boundary layer, only u'w' is generallynon-zero. However, when turbulence containsmotions that are both spatially and temporallyorganized; i.e., there is a definite phase structurebetween the velocity components, all three meanReynolds stress components can be finite.Instantaneously, all three components can take onlarge positive or negative values. The passage of anorganized patch of turbulence can be identified usingan anemometer capable of the rapid, simultaneousmeasurement of multi-axis wind components.Coherent turbulent structures can contain largevelocity shears over small distances as well assignificant local vorticity. Moderate to high-speedflows that are just slightly or weakly stable (Ri small,but positive) often contain small, energetic turbulencestructures that, when ingested by wind turbine rotors,increase fatigue damage.2

The examination of many load histories with thecorresponding three instantaneous Reynolds stress

components has supported the conclusions reachedfrom the statistical analyses; i.e., significant loadingevents are often related to the ingestion of coherentturbulent structures. Figure 1 demonstrates thisconnection. Here the actual zero-mean time historyof a turbine rotor flapwise load and the correspondinginstantaneous Reynolds stress components, derivedfrom a multi-axis sonic anemometer mounted on atower upstream of the turbine, are presented.Because the turbulence measurement is upstreamfrom the rotor, the stress/load time relationships maynot be accurate because of the propagation delay. Itis also possible that, because of a skewed flow, theindicated turbulence patch may not been seen by theturbine and vice versa.

Figure 1 clearly demonstrates the issue. We arevery much interested in the details of the turbulentinflow responsible for the tensile and compressionloading peaks. The loading extremes residing withinthese peaks define the stress cycles seen in the highamplitude tails of loading spectral distributions. Wewould like to be able to identify the characteristics ofthe turbulent structure, which through the action ofthe turbulence/rotor interaction, produce the observedunderstand the atmospheric conditions that produce

Hub Reynolds stress components

Time (s)0 25 50 75 100 125 150

(m/s

)2

-40

-20

0

20

40

Zero-mean root flapwise bending

kNm

-10

-5

0

5

10

u'w'u'v'v'w'

3

Figure 2. Example of conventional power spectrum of non-stationary flapwise time history

aeroelastic response. Further, we would like tounderstand the atmospheric conditions that producesuch structures and to be able to numerically simulatethem. Finally, we wish to understand the details ofthe aerodynamic/aeroelastic processes involved inorder to improve our unsteady aerodynamics modelsand to use those models as a tool in developing loadmitigation strategies.

The time-series of Figure 1 represent examplesof non-stationary processes. The turbulent bursts inthe Reynolds stress field can produce a short-livedaeroelastic response in the root loads of the turbinerotor. In analyzing this response, we wish to examinethe excitation and response in the frequency domain.Conventional spectral techniques are based on timehistories being at least "quasi-stationary", i.e. , thestatistics of the signal are not changing with time orat least changing very slowly over the period of therecord. Clearly, that is not the case with any of the

signals depicted in Figure 1. The use of conventionalspectral analysis of a flapwise loading signal that ischanging characteristics until the influence a short-lived turbulent burst is demonstrated in Figure 2.The spectrum of this signal has excellent frequencyresolution (0.1 Hz). However, one cannot tell whatthe spectral energy distribution is in the 0 to 5 and 35to 60 second portions of the record relative to thelarge excursions seen between 5 and 35 seconds. Wesay the spectrum of Figure 2 has very good frequencyresolution or localization but very poor timeresolution or localization. In fact, the timelocalization is the record length itself.

We wish to identify suitable techniques to allowus to dissect important frequency domain informationfrom loading events that typically last only on theorder of 10 seconds or less. We would like to usethat information to understand the nature of theturbulence field excitation and the forced responses

Frequency (Hz)

0.1 1 10

Ro

ot

flap

load

(kN

m)2 /H

z

10-5

10-4

10-3

10-2

10-1

100

101

102

103

Zero-mean flapwise loads

Time (s)0 10 20 30 40 50 60

kNm

-15-10

-50

51015

20

1-P

4

of the turbine rotor that result in large load excursionsand significant fatigue damage. In this paper, wediscuss our initial progress in developing amethodology to allow us to achieve these objectives.

APPROACHWe need to identify an analytical tool or

collection of tools that allow us to perform a "local"analysis of transient loading events seen in theturbine rotor loading histories and their associatedturbulent inflow structures. Conventional Fourierspectral analysis does not meet this criteria. Ourapproach is to evaluate the capabilities of what isgenerally known as Time-Frequency Analysis and itssub-discipline, Wavelet Analysis. We examined bothobserved and simulated turbine inflows and resultingaeroelastic responses afforded us by these analysistechniques.

Specifically, we report on our progress in

• Identifying the most promising techniquesavailable in the analysis of transient signals thatcan be applied to the excitation and response ofwind turbine rotors

• Analyzing both the inflow Reynolds stress androot flapwise rotor response time historiesderived from field measurements from arelatively rigid turbine design (Micon 65) and avery flexible one (Wind Eagle CWE300) usingseveral of the identified techniques

• Analyzing the results of a numerical simulationof the very flexible turbine design includingparameters measured in the rotational frame notavailable from actual observations

• Reaching initial conclusions about theapplicability of these techniques tounderstanding the turbulence/rotor interactionand its role in transient load events.

AN EVALUATION OF TIME-FREQUENCYANALYSIS TOOLS

Our goal in this study is to evaluate various time-frequency and wavelet analysis techniques that allowus to obtain an estimate of the frequency content of asignal locally at a particular point in time. This waywe can assess the impact of short-lived inflowtransients on the aeroelastic response of a windturbine rotor and their relationship to large stresscycles and fatigue accumulation.

Linear time-frequency transforms correlate thesignal being analyzed with a family of waveformsthat are well-localized in time and frequency.3 Twoof the most important classes of these families are theWindowed Fourier and wavelet transforms. In our

study we evaluated examples of both classes usingwind turbine related signals such as the instantaneousshearing stresses and flapwise load signals. Weapplied the Windowed Fourier transform (sometimesreferred to as the Short Time Fourier Transform orSTFT) through an analysis procedure called thespectrogram using various window functions. Itmeasures the energy in a time-frequencyneighborhood, fi,ti specified by a resolution box(called a Heisenberg box) bounded by σfσt. Weapplied the same procedure using the wavelettransform (now called a scalogram) and variousanalyzing wavelet functions. While each approachhas its advantages and disadvantages, we found that(at least for a constant-speed turbine) thesetechniques did not offer us any additional informationover the continuous and discrete wavelet transformsdiscussed below. We now provide a brief discussionof the theory behind the specific analysis techniquesemployed in this research.

Wavelets are a powerful tool for studying time-frequency behavior of finite-energy signals. Theadvantages of the wavelet decomposition overFourier transforms present themselves whentransforming a signal that has time varying and/orsingularity characteristics. Multiresolution andwavelet theory has recently found applications in aremarkable range of disciplines such as datacompression, signal processing, image analysis,statistics, and modeling of nonlinear dynamicprocesses. An excellent presentation of the subject isgiven by Mallat.3 Note: It is not the intention of thisshort section to give a broad introduction to thesubject or a long list of references. In this research,we apply wavelet analysis to establish therelationship between the turbulent wind and themechanical loads in the blades of the wind turbine.We use both discrete and continuous wavelettransforms to obtain the best possible presentation ofthe mechanical loads of the relatively stiff blades ofthe Micon 65 turbine and the very flexible blades ofthe CWE300 machine. Since wavelet analysis ofsignals is a new technique we start with its briefpresentation.

The Continuous Wavelet Transform (CWT)The continuous wavelet transform Wf(s,b) of a

finite-energy signal f(t) is defined as its scalarproduct with the wavelet ψsb(t). In other words, thewavelet transform represents the correlation of thesignal f(t) and the wavelet ψsb(t). In addition, due toParseval’s theorem, this relationship can also bestated in the frequency domain. The followingequation gives both representations:

5

∫

∫∞

∞−

∗

∞

∞−

=

=

ωωψωπ

ψ

ω desf

dttftbsWf

jb

sb

)(ˆ)(ˆ21

)()(),(

(1)

where the wavelet

RR ∈∈−= +−− bsbtsstsb , , ))(()( 12/1 ψψ (2) (2)

and ψ(t), known as the mother wavelet, is a function,which satisfies the following equation:

0)( =∫∞

∞−

dttψ . (3) (3)

One such function, which we used in ourresearch, is the Morlet wavelet, shown in Figure 3.We chose this analyzing wavelet because it issymmetrical and exhibits a linear phase response. Itcan be seen from equation (2) that the parameters sand b in equation (1) can be interpreted as the scaleand the translation. The mother wavelet correspondsto scale s =1. The increase of s is equivalent tostretching the mother wavelet or moving itsfrequency content towards lower frequencies. On thecontrary, the decrease of s causes the compression ofthe mother wavelet or moving its frequency contenttowards higher frequencies. Varying b for the fixedscale corresponds to sliding a wavelet with a fixedbandwidth and with some center frequency along theanalyzed signal. For a given position b of theanalyzing wavelet, we calculate the value of thewavelet transform. It tells us, to what extent thefrequency content of the analyzed signal, in theneighborhood of b, is close to the frequency contentof the wavelet at the given scale s.

To make the described procedure useful, wehave to establish the relationship between the scale sand the frequency content, at this scale, of aparticular analyzing wavelet. This is accomplishedby defining two frequency characteristics of themother wavelet1, the center frequency

∫∞

=0

20 )(ˆ

21

ωωψωπ

ω d , (4)

and the bandwidth B0=σω, centered around ω0, with

∫∞

−=0

220

2 )(ˆ)(21 ωωψωωπ

σ ω d . (5)

For a wavelet of the scale s, the center frequency andthe bandwidth are, respectively, defined by thefollowing equations:

Figure 3. The Morlet wavelet

, , minmax00

ssB

Bs ss

ωωωω

−=== (6)

with 2/ , 2/ 00min00max BB −=+= ωωωω . Note

that by this convention 10 BB = and 10 ωω =It is very important to note that both the center

frequency and the bandwidth have different valuesfor every wavelet. In other words, it is important tochoose the mother wavelet with high frequencyresolution or the narrow bandwidth. The increase ofthe scale s improves the frequency resolution at lowerfrequencies.

In practice, for both continuous-time anddiscrete-time wavelet signal analysis, we have to usethe discrete-time representation of the analyzedsignal. This signal is sampled with some frequency

Tω , where T is the sampling period. In this case, the

bandwidth of the sampled signal is limited by theNyquist frequency 1/2 Tω . The frequency variation

of the continuous-time signal between 0 and1/2 Tω corresponds to the representative point

moving between these values in the complex plane salong imaginary axis. However, the correspondingfrequency variation of the discrete signal should berepresented in the complex plane z. The mappingbetween the imaginary axis of the s plane, for whichs=jω is determined by the following equation:

1 ,or ==∠= zTzez ti ωω ,

which means that when the frequency of thecontinuous-time signal varies in the interval [0,1/2 Tω ], the angle or the frequency of the discrete-

time signal, represented in radian units, varies in theinterval [0,π]. This is justified by the followingcalculation:

ω/π

( )tψ

6

πω

πωω ==∠

T

TT

z2

22/ .

In other words the frequency of 1/2 Tω rad/s of

the continuous-time signal is represented by thefrequency of π radians of the discrete-time signal.This justifies the use of the relative frequency of thediscrete-time signal, which is obtained by dividing,by π, every frequency ω from the interval [0,π].Then, the relative frequency ω/π=1 corresponds tothe highest frequency of the continuous-time signalpreserved in the sampled signal, i.e., the frequency1/2 Tω rad/s or the frequency 1/2 Tf Hz, with

TT fπω 2= . It is convenient to represent the

magnitude of the Fourier transform of the waveletswith different scales as a function of the relativefrequency. Then, we obtain the real frequency valuemultiplying the relative frequency by the samplingfrequency represented in radians per second or inHertz.

The magnitude of the Fourier transform of theMorlet wavelet for several scales s (s=1,2,3,4,5) isrepresented in Figure 4. The Morlet mother waveletis given by the following equation:

)5cos()( 2/2tet t−=ψ .

Figure 4. Magnitude of the Fourier transformof Morlet wavelet

For the Morlet wavelet ω0=5 and B0=1.701(radian units). Using these two parameters, we candetermine ωs and Bs for any value of s. It is clearfrom Figure 4 that the magnitude of the Fouriertransform of the Morlet wavelet for s=1 is almostentirely positioned to the right of ω/π=1 or to theright of the Nyquist frequency. In other words, it isalmost completely uncorrelated with the discreterepresentation of the original signal or the usefulbandwidth of the Morlet mother wavelet is almost

entirely above the half of the sampling frequency. Itis apparent from this figure that the frequencyresolution is much better for the lower relativefrequencies (or for the lower scales). Thisimmediately offers a solution to the problem of poorresolution at higher frequencies, i.e., the analyzedsignal can be over-sampled at such a rate that lowerscales and, in particular, scale equal one will notcontain any useful frequencies. In other words, thebands of the lower scale wavelets (or wavelets withfrequency resolutions too small for the performedanalysis) will be outside the essential bandwidth ofthe analyzed signal.

It turns out that both the center frequency and thebandwidth of the wavelet, represented as a functionof the scale s on the log-log scale, are linear andprovide direct scale to frequency conversion. This isvery important for the understanding and theinterpretation of the results of the wavelet analysis ofsignals.

In the data acquisition for the CWE300 turbine,we used a pre-sampling filter with a cut-offfrequency of 40 Hz and a sampling rate of 200/s, i.e. ,we over-sampled the signals five times. We initiallyobtained very good frequency resolution in therelatively low-frequency bandwidth of interest.However, for the Micon 65, the pre-sampling filterand sampling rate used were 13 Hz and 32/srespectively. Further, the sampling rate availablefrom the numerical simulation of the CWE300turbine was only 60/s. Clearly the temporalresolution of the original Micon 65 signals wereinadequate to achieve good frequency resolution overthe range of interest and the simulated results could

Figure 5. Continuous transform scale-to-frequency conversion/bandwidth for Morlet

wavelet at 240/s sampling frequency

ω/π

( )|ˆ| fψ

CWT Scale (s)

6 8 15 20 30 40 60 80 150 200 300 40010 100

Sca

le c

ente

r fr

equ

ency

an

d b

and

wid

th (

Hz)

0.060.08

0.2

0.4

0.60.8

2

4

68

20

40

0.1

1

10

center frequencybandwidth

7

be improved. In order to accommodate theserequirements, we over-sampled each by linearinterpolation to achieve a sampling rate of 240/s.The scale-to-frequency conversion characteristics forthe Morlet wavelet and this sampling rate are shownin Figure 5. It should noted, as can be seen in Figure4, that the individual wavelet passbands do havesome overlap that, in effect, reduces the frequencylocalization or discrimination.

The Discrete Wavelet Transform (DWT)We shall now discuss the discrete wavelet

transform. As a result of the quantization of theparameters s and b, we obtain the following equation,which corresponds to equation (2)

2),(002/

0 )()(Z∈

−− −=nm

mmmn nbtaat ψψ . (7)

In particular, we can choose 1 and 2 00 == baand obtain the following dyadic orthonormal waveletbasis of the L2(R) space:

))2(2(2)( 2/ ntt mmmmn −= −− ψψ . (8)

Note that for a fixed value of m we have a constantscale 2m. Using )(0 tmψ as a basic wavelet at the

level m, we can rewrite (8) as

)2(0 nt mmmn −=ψψ , (9)

which means that 2m is the time step of shifting thewavelet )(0 tmψ . In the discrete wavelet analysis of

the signal f(t)∈L2 we determine the inner products ofthe signal f(t) and the sequence of functions ψmn.These inner products are known as waveletcoefficients. The discrete wavelet expansion of thesignal f, which is the inverse discrete wavelettransform, can be expressed as

mnm n

mnm

mnmn ndftf ψψψ ∑ ∑∑ == ][),()(,

, (10)

where the wavelet coefficients dm[n] = (f,ψmn)represent the common features of the signal f andwavelet ψmn. The parameters m and n allow to accessthe particular signal features. The parameter n isused to localize the requested time instant and theparameter m allows to choose the scale level orfrequency range , in which we wish to examine thesignal frequency spectrum. Let us now concentrateon the interpretation of (10). For a fixed value of m,the right-hand side of this equation represents anorthogonal projection of f onto a subspace

Wm⊂L2(R). Denoting this projection by fPmW , we

can represent (10) as

∑∑∞

+=

=

−∞=

+=1

Jm

WW

Jm

m

fPfPfmm

. (11)

We can treat the second sum as an orthogonalprojection fP

JV of f onto certain subspace

VJ⊂L2(R). We can rewrite (11) as

fPfPfJm VW

Jm

m

+= ∑=

−∞=

. (12)

This leads to the following relation:

Z∈⊕= ++ jWVV jjj , 11 , (13)

which means that Vj+1 is the orthogonal complementof Wj+1 in Vj.

If the set });({ Z∈− nntϕ is an orthonormal basis

in V0, then we call ϕ the scaling function and wedefine an orthonormal dyadic base in Vj as

Z∈−= −− nntt jjjn , )2(2)( 2/ ϕϕ (14)

then we can rewrite (12) as

Jnn

J

J

j nj nandtf ϕ∑∑ ∑

∈−∞= ∈

+=ZZ

][][)( , (15)

where ),(][ JnJ fna ϕ= and is known as the

approximation of the signal f at the level J . This isthe celebrated multiresolution representation of thefunction f.

We can approximate the signal f by its projectiononto some subspace Vl and then, we obtain

Jnn

J

J

lj nj nandtf ϕ∑∑∑

∈= ∈

+≅ZZ

][][)( . (16)

Equation (16) is an approximate multiresolutionrepresentation of the signal f(t). To make thismultiresolution representation complete, let us notethat

}.{

)(2

∅=⊃⊃⊃⊃⊃⊃= ∞∞− VVVVL Jl KKKR

(17)

In each step of the process of the multiresolutionanalysis, we obtain the detail of a signal, whichrepresents the projection of its previousapproximation onto a subspace Wj with the subscriptincreased by 1 and the new approximation, whichrepresents the projection of the previousapproximation onto the subspace Vj with the subscriptincreased by one. Finally, when we stop the process,the signal representation consists of a sequence ofdetails and the last approximation. In the next

8

section, we only present a subset of the details. Wechose the 8th-order Symmlet as the analyzing waveletfor the multiresolution analysis. High-order Symmletwavelets achieve the closest approximation of exactsymmetry and linear phase of the wavelet familiesthat are useful with the discrete transform. Wedesired to minimize the phase distortion in order tomaintain the temporal relationships between signalsin the detail band as close as possible. The Fouriertransform of the 8th-order Symmlet wavelet is plottedin Figure 6. It can be seen that the passbands arewider than the Morlet in Figure 4.

Figure. 6 Magnitude of the Fourier transform ofthe 8th-order Symmlet wavelet

WAVELET ANALYSIS OF RIGID ANDFLEXIBLE TURBINE DESIGNS

We analyzed observed representative turbulenceand flap load signals from both rigid and veryflexible turbines using both the continuous anddiscrete wavelet transforms (for multiresolutionanalysis). The inflow measurements were limited toa single point at or near hub height upstream of theturbines. We used the Matlab Wavelet Toolbox asthe primary analysis software for this research.

Rigid TurbineWe analyzed the turbulence excitation and rotoraerolelastic response of an upwind, three-bladedMicon 65/13 turbine located in San Gorgonio Pass ofCalifornia. The instantaneous Reynolds or shearingstress components of the inflow were measured atabout hub-height and 1.5-rotor diameters upstream. Amulti-axis sonic anemometer with a measurementbandwidth of 5 Hz and a sampling rate of 16/s wasused. The response variable is the root flap-bendingload measured on one of the blades. The load signalwas sampled at 32/s but a pre-sampling filtereffectively limited the bandwidth to about 13 Hz.

The blade was an NREL Thin Airfoil5 prototypeconsisting of S806A, S805A, and S807 airfoil shapes.The turbine rotor was 17 m in diameter, a hub heightof 23 m, and turned at 48 rpm. It was located at Row37 within a 41-row wind farm.

An example of the analysis is presented inFigures 7, 8, and 9. We analyzed a record length of60 s using both the continuous wavelet transform andmultiresolution analysis (discrete transform) in orderto more closely examine the loading transients andtheir excitation presented earlier in Figure 2. In orderto achieve greater frequency resolution at the higherfrequencies with the wavelet transforms, we over-sampled the original signals to achieve a finalsampling rate of 240/s by linear interpolation. Thecontinuous transform, using the Morlet analyzingwavelet, was calculated for a range of scales in stepsof 5 between 5 and 575. Using the attributes of theMorlet wavelet, a scale value of 5 corresponds to acenter frequency of 31.197 and bandwidth of 6.497Hz. For a scale of 575, the correspondingfrequencies were 0.332 and 0.056 Hz, respectively.

The multiresolution analysis using the discretewavelet transform was calculated using an 8th-orderSymmlet analyzing wavelet to nine levels of detail.The first three detail bands (with frequency ranges of60-120 Hz, 30-60 Hz, and 15-30 Hz) were ignored astheir minimum frequency exceeded the effective 13-Hz limitation of the original signals. The remainingsix detail bands are referred to as B1 to B6. Table 1summarizes the frequency ranges of each detail bandand characteristic turbine modal responses that residewithin each.

An analysis summary of the full 60-secondrecord is presented in Figure 7. The time histories ofthe hub-height wind speed, Reynolds-stresscomponents, and flap load signals are placed at thetop. Immediately below the time histories is thecontinuous wavelet transform analysis of the flapwisesignal and its corresponding multiresolution analysisbelow that.

The dark areas on the continuous transformdisplay represent frequency bands containing highvalues of stress energy. The darker the band, thehigher the dynamic stresses at the blade root. Thereis considerable low-frequency stress energy in the0.33- to 0.7 Hz frequency range (0.4 to 0.9 P, whereP is the blade passage frequency of 0.8 Hz) for thefirst 27 s. A strong broadband response is associatedwith the load peak between 12 and 18 s; i.e.,considerable stress energy extending up to10 Hz(~13P). A similar, though not as intense, response isassociated with the load peak between 30 and 32 s.The broad period of large flap amplitude activity

ω/π

( ) |ˆ| fψ

9

Table 1. Multiresolution Analysis Detail Frequency Ranges for Micon 65 Turbine

DetailBand

CyclicFrequencyRange (Hz)

Characteristic Modal Responses

B1 7.5 - 15.0 Rotor 2nd flap bending; 2nd asymmetric flap bendingB2 3.75 - 7.5 Rotor 1st lag bending; 2nd asymmetric lag bendingB3 1.875 - 3.75 Rotor 1st symmetric flap bending, 1st/2nd asymmetric flap bending; tower fore/aft

and side/side asymmetric bendingB4 0.938 - 1.875 Tower 1st fore/aft and side/side bendingB5 0.469 - 0.938 1-PB6 0.234 - 0.469

from about 5 to 32 s is dominated by frequencies inthe B4 and B5 detail bands (tower motions). Thesebands include the blade passage frequency (1-P) andthe 1st-tower modes. A careful examination of thepeak response between 12 and 18 s shows strongresponses in detail bands B2, B3, B4, and B5 of themultiresolution analysis that correspond to thebroadband response indicated by the continuoustransform above. Thus during this load peak, themodal responses within the frequency range definedby the detail bands were simultaneously or coherentlyexcited.

The first 20 s of the record is in Figure 8 usingthe same format as Figure 7. It provides a closerexamination of the record. The analysis describes theevolution of the flap load signal leading up to thelarge excursions between 14 and about 18 s. Thecharacter of the flap signal changes simultaneouslywith the peaks in the inflow Reynolds stresses atabout 5 s. During this period, there is significantresponse in the B4 and B5 detail bands (0.47 to 1.88Hz or 0.6 to 2.3P) with occasional bursts into the B3band (2.3 to 4.7P). Many of the lower rotor modesare located in the B3 band (see Table 1). Thesignificant broadband response occurring between 14and 15 s appears to be associated with the increasedlevels of coherent turbulence activity as indicated bythe Reynolds stress time histories.

The details of the inflow turbulence (Reynoldsstress) and rotor flap response are plotted in Figure 9.Here, in addition to the time histories of the Reynoldsstress components, the continuous wavelet transformsof each stress component and the flap signal arepresented. The stress component diagrams indicatethat there is considerable broadband energyassociated with these bursts of coherent turbulencethat are being ingested by the turbine rotor. Itappears that the first burst near 5 s starts the rotor andtower oscillating at the various lower modalfrequencies. The second burst, beginning about 12 s

and lasting until 15 s, is responsible for adding theadditional energy to cause the broadband loadingpeak between 14 and 15 s. The equivalent turbulentwavelengths for the Reynolds stresses are the orderof 1 to 4 meters or 6% to 23% of the rotor diameter.Thus coherent turbulence structures whose spacescales are less than a quarter of the rotor diameter canplay an important role in peak load responses.

Flexible TurbineWe analyzed the turbulence excitation and rotor

aeroelastic response of a flexible, two-bladed,downwind turbine. The turbine was a Carter/CannonWind Eagle CWE300 pre-prototype5,6 installed at theNational Wind Technology Center (NWTC). Theinstantaneous Reynolds stresses of the inflow aremeasured two rotor diameters upstream using amulti-axis sonic anemometer with a measurementbandwidth of 10 Hz, a 12.5-Hz pre-sampling filter,and a sampling rate of 200/s. The response variableis the flapwise bending measured near the hub on theflexible inner spar (flexbeam) of the turbine rotor.This signal was passed through a 40-Hz pre-samplingfilter and sampled at 200/s. The turbine rotor was29.3 m in diameter, a 48.8 m hub height, and turnedat 57 rpm. The turbine was operated with no othersupstream. The segment analyzed came from a longer10-minute record in which the hub-height averagewind speed was 18.33 m/s and the flow was slightlystable with a turbine layer Richardson number valueof +0.031. To better compare with the numericalsimulation of this record, the bandwidth of theturbine signals prior to analysis was low-pass limitedto 30 Hz.

Again, a 60-s record was analyzed and the resultsare presented in Figure 10 in the same format asFigures 8 and 9. As previously, the sampling ratewas increased to 240/s by linear interpolation toproduce an identical scale range for the continuoustransform using the Morlet analyzing wavelet.

10

B1

B

2

B3

B

4

B5

B6

Root flap load

Cyc

lic f

requ

ency

(H

z)

time (s)0 10 20 30 40 50 60

1-P

0.4

0.5

0.7

0.6

0.8

1.01.21.5

3.05.0

10.0

2.0

kNm

Horizontal wind speed

m/s

68

1012141618

20

Reynolds stress components

(m/s

)2

-40

-20

0

20

40

Zero-mean flapwise loads

kNm

-15-10

-505

101520

Figure 7. Time and wavelet analysis of rigid turbine turbulence response

11

Horizontal wind speed

m/s

68

101214161820

Reynolds stress components

(m/s

) 2

-40

-20

0

20

40

Zero-mean flapwise loads

kNm

-15-10

-505

101520

time (s)0 5 10 15 20

B1

B

2

B3

B

4

B5

B6

Root flap load

Cyc

lic f

requ

ency

(H

z)

1-P

0.4

0.5

0.7

0.6

0.8

1.01.21.5

3.05 . 0

10.0

2 . 0

kNm

Figure 8. Time and wavelet analysis of detail of rigid turbine response

12

time (s)0 5 10 15 20

u’w’ component

u’v’ component

v’w’ component

Root flap load

Cyc

lic f

req

uen

cy (

Hz)

0.7

0.8

1.0

1.2

1.5

2.0

3.05.0

10.0

0.7

0.8

1.0

1.2

1.5

2.0

3.05.0

10.0

0.7

0.8

1.0

1.2

1.5

2.0

3.05.0

10.0

1-P

0.4

0.5

0.7

0.6

0.8

1.01.21.5

3.05.0

10.0

2.0

Reynolds stress components

(m/s

)2

-40

-20

0

20

40

Figure 9. Continuous wavelet analysis of inflow and flap response for rigid turbine detail

13

Table 2. Multiresolution Analysis Detail Frequency Ranges for CWE300 Turbine

DetailBand

CyclicFrequencyRange (Hz)

Characteristic Modal Responses

B1 15.0 - 30.0 Rotor 1st/2nd torsion bending; 3rd symmetric lag bendingB2 7.5 - 15.0 Flexbeam 2nd flap bending; blade shell 4th flap bendingB3 3.75 - 7.5 Rotor 3rd symmetric and asymmetric bending; 2nd asymmetric lag bending; blade

shell 2nd flap bendingB4 1.875 - 3.75 Rotor 2nd asymmetric flap bending; blade shell 1st flap bendingB5 0.938 - 1.875 Rotor 1st asymmetric flap bending; rotor 2nd symmetric flap bending; tower 1st/2nd

fore/aft and side/side bending; drive train 1st bending; blade shell 1st flap bendingB6 0.469 - 0.938 Rotor 1st asymmetric lag bending; 1-PB7 0.234 - 0.469 Rotor 1st symmetric flap bending

As previously, the sampling rate was increased to240/s by linear interpolation to produce an identicalscale range for the continuous transform using theMorlet analyzing wavelet. As before, an 8th-orderSymmlet wavelet was applied to compute themultiresolution analysis with the discrete transformof the signals. An additional detail band wasincluded (15-30 Hz) because of the wider signalbandwidth associated with this turbine. Thefrequency ranges for each detail band and thecharacteristic modal responses are listed in Table 2.A comparison of the contents of Tables 1 and 2 willreveal the much lower frequencies associated withmany of the flexible-turbine mode shapes relative tothose seen in the much more rigid design.

The record from the flexible turbine has twoperiods of oscillating loads. The oscillation in thefirst 12 s appears to be associated with a broad peakin the wind speed that contains a very complexturbulent structure. The second oscillation envelopeappears to be in response to the turbulent structureseen at about 30 s into the record. As previously, thefirst 20 seconds is detailed in Figure 11. Here it isclear that the large oscillation is related to theturbulent structure between 2 and 11 s. This is madeeven more obvious by the continuous transforms ofthe individual shear stress components and theirrelationship to the flapwise load response presentedin Figure 12. From these graphs and those from theMicon 65, one of the major consequences of the rotorencountering these organized turbulent regions is thatthe higher-order mode shapes are energized in aphase coherent manner. The organized inflowexcitation results in large load excursions when thepeaks in the constituent modal responses come intophase alignment. This can be seen by examining thesignals in detail bands B2 through B5 in themultiresolution graphs at the bottom of Figures 8 and11.

WAVELET ANALYSIS OF A SIMULATEDFLEXIBLE TURBINE RESPONSE

One of the major obstacles in interpreting therole of coherent turbulence regions and their impacton a turbine rotor when using normally availableobserved data is that the excitation is usually onlymeasured at fixed locations in space. The rotor,however, is continually passing through a muchlarger area and rotationally sampling the turbulentfield within that area. To obtain at least an estimateof what a rotating blade may encounter as it passesthrough a region of coherent turbulence, we used anumerical simulation of the CWE300 turbinedeveloped by Wright.6 The inflow boundaryconditions for the simulation were chosen to be asclose as possible to those associated with thecollected data presented in the previous section. Aspecial model calculation was made in which timehistories of a range of aerodynamic parameters(angle-of-attack, normal and tangential forces, andlift and drag coefficients) for each of the outer sevenblade sections was made available in rotational space.In addition to the aerodynamic parameters, therotationally sampled, three wind components presentat each blade section were also output. The turbulentinflow to the model was generated by theSNLWIND-3D code.7 This code has the ability toproduce complex, turbulent structures similar to thoseseen in the observed data discussed previously.

The loads on the turbine during the actualobservations were often quite high due to the strongand very turbulent winds in which the turbine wasoperating. Gusts exceeding 27 m/s were seen duringthe 10 minutes of actual data collection. Weexamined the simulated record for an example of alarge flap load excursion in order to study thecharacteristics of the inflow in both fixed androtationally sampled space. Figure 13 displays thetime and wavelet analysis of a 20-second record in

14

time (s)0 10 20 30 40 50 60

Cyc

lic fr

eque

ncy

(Hz)

B1

B

2

B3

B

4

B5

B

6

B7

Root flap load0.4

0.5

0.7

0.6

0.8

1.01.21.5

3.05 . 0

10.0

2.0

1-P

kNm

Reynolds stress components

(m/s

)2

-40-30

-20

-100

10

2030

40

Zero-mean flapwise load

kNm

-10

-5

0

5

10

Horizontal wind speed

m/s

12

16

20

24

28

Figure 10. Time and wavelet analysis of detail of flexible turbine turbulence response

15

time (s)0 5 10 15 20

Cyc

lic fr

eque

ncy

(Hz)

H o r i z o n t a l w i n d s p e e d

m/s

1 2

1 6

2 0

2 4

2 8

R e y n o l d s s t r e s s c o m p o n e n t s

(m/s

) 2

- 4 0

- 2 0

0

2 0

4 0

Z e r o - m e a n f l a p w i s e l o a d s

kNm

- 1 0

-5

0

5

1 0

kNm

B1

B

2

B3

B

4

B5

B

6

B7

Root flap load

1-P

0.4

0.5

0 . 7

0 . 6

0 . 8

1 . 01.21.5

3.05.0

10.0

2.0

Figure 11. Time and wavelet analysis of detail of flexible turbine response

16

Reynolds stress components

(m/s

)2

-40

-20

0

20

40

u’w’ component0 . 4

0 . 5

0 . 7

0 . 6

0 . 8

1 . 01 . 21 . 5

3 . 05 . 0

1 0 . 0

2 . 0

Cyc

lic fr

eque

ncy

(Hz)

u’v’ component

v’w’ component

0 . 4

0 . 5

0 . 7

0 . 6

0 . 8

1 . 01 . 21 . 5

3 . 05 . 0

1 0 . 0

2 . 0

0 . 4

0 . 5

0 . 7

0 . 6

0 . 8

1 . 01 . 21 . 5

3 . 05 . 0

1 0 . 0

2 . 0

0.4

0.5

0.7

0.6

0.8

1.01.21.5

3.05.0

10.0

2.0

Root flap load

1-P

time (s)0 5 10 15 20

Figure 12. Continuous wavelet transform analysis of turbulence excitation for flexible turbine

17

which large flap excursions were predicted. As in theobserved data, a broad period of high winds containsa very complex turbulent structure. The high loadsare confined to four blade passages through thecomplex turbulence region between 7 and 12 s.Again, there is a broadband response to this turbulentregion as is seen in both the continuous andmultiresolution wavelet analyses in the lower portionof Figure 13.

In Figure 14, we compare the turbulenceexcitation and response in both fixed and rotationalspace. In the two upper panels, we compare theinstantaneous Reynolds stress components as seen infixed space at the rotor hub with those seen by the78%-span blade station as it rotates through theturbine inflow area. The rotational picture is quitedifferent from that in fixed space though the presenceof complex turbulence structures can be identified.The continuous wavelet transform analysis of thenormal force predicted for the 97%-span station ispresented in the third panel. The analysis shows thatthere is substantial energy concentrated in the 0.9 to 7Hz (0.95 to 7 P) range and possibly above during theperiods of maximum excitation. This is thefrequency range in which the first and secondsymmetric and asymmetric rotor mode shapes reside.These results suggest that one of the consequences ofa blade passing through a coherent, turbulent fieldwill be the excitation of the first and higher ordersymmetric and asymmetric rotor modes. This resultis consistent with what Tangler et al.8 found inanalyzing the Micon 65 data. It was noted thatcoherent turbulence excited the first asymmetricmode of the blade that contributed to large flapwiseresponses

CONCLUSIONSWe applied both the Windowed Fourier and wavelettransforms to both inflow turbulence (Reynoldsstresses) and a turbine key rotor response parameter(root flap loads). We found, while each approach hasits advantages and disadvantages, the WindowedFourier or Short-Time Fourier Transform did notoffer us any additional information over what wasavailable from the continuous and discrete wavelettransforms. Since the turbines we analyzed wereboth of constant-speed design, this conclusion maynot extend to machines operating in a variable speedmode. As expected, with proper signal preparation,i.e., over-sampling, both the continuous and discretewavelet transforms provided insights into theturbulence/rotor interaction that could not beachieved from conventional spectral analysis.

Further, our initial application of the wavelet analysistechniques for interpreting the turbulence/rotorinteraction in the rotating frame derived fromnumerical simulations appears quite promising.

Our analysis of the both the rigid and flexibleturbines has shown that a coherent or broadbandstructural response occurs in a rotor ingesting acoherent patch of turbulence. The analysis of theinflow revealed that such patches contain a broadrange of frequencies (turbulent eddy wavelengths)that contribute to the excitation of a wide range ofmodal responses in the turbine rotor and supportingstructure. The multiresolution analyses of the rotorflapwise responses revealed that the load peaks occurwhen the various modal frequencies also reach peakvalues in phase or unison. The first and secondsymmetric and asymmetric rotor modes appeared tobe most susceptible to such excitation. It was foundthat the constituent turbulent eddies of coherentstructures, whose equivalent space scales are lessthan a quarter of the rotor diameter, play a major rolein developing peak load responses.

Our results indicate that the presence of coherentturbulent structures, as revealed by the inflowReynolds stress field, is a major contributor to largeload excursions and the accompanying fatiguedamage. We have demonstrated that wind gusts cancontain a very complex turbulence structure that inturn can provide broadband excitation to a turbinerotor passing through it. Thus when assessing apotential wind energy site or simulating turbulentinflows for load assessments on new turbine designs,the frequency and intensity of such flow structuresneeds to be evaluated in order to assess potentialfatigue damage accumulation.

FUTURE WORKWe plan to continue the analysis of the

turbulence/rotor interaction using numericalsimulations and, if available, observed data derivedfrom the Unsteady Aerodynamics Experiment beingconducted by the National Wind Technology Center.We also wish to apply these techniques to evaluatingthe load response characteristics from variable speedrotors.

ACKNOWLEDGEMENTSThe authors wish to acknowledge the

contributions of Alan Wright and Kirk Pierce ofNREL to this project.

18

B1

B

2

B3

B

4

B5

B

6

B7

time (s)0 2 4 6 8 10 12 14 16 18 20

Root flap load0.4

0.5

0.7

0.6

0.8

1.01.21.5

3.05.0

10.0

2.0Cyc

lic fr

eque

ncy

(Hz)

1-P

H o r i z o n t a l w i n d s p e e d

m/s

1 2

1 6

2 0

2 4

2 8

3 2

R e y n o l d s s t r e s s c o m p o n e n t s

(m/s

) 2

- 8 0- 6 0- 4 0- 2 0

02 04 06 0

Z e r o - m e a n f l a p w i s e l o a d s

kNm

- 1 5

- 1 0

-5

0

5

1 0

kNm

Figure 13. Time and wavelet analysis of simulated turbulence/rotor interaction

19

Figure 14. Time and continuous wavelet analyses of fixed and rotating excitation and response

Hub Reynolds stress components(m

/s)2

-80-60-40-20

0204060

Normal force @ 97% span

Zero-mean flapwise loads

time (s)0 2 4 6 8 10 12 14 16 18 20

kNm

-15

-10

-5

0

5

10

0.4

0.5

0.7

0.6

0.8

1.01.21.5

3.05.0

10.0

2.0

Cyc

lic f

req

uen

cy (

Hz)

1-P

(m/s

) 2

-40-30

-20-10

0102030

Rotationally-sampled Reynolds stresses @ 78% span

20

REFERENCES1. Kelley, N.D. "The Identification of Inflow FluidDynamics Parameters That Can Be Used to ScaleFatigue Loading Spectra of Wind Turbine SpectralComponents," NREL/TP-442-6008, Golden, CO:National Renewable Energy Laboratory, 1993.

2. Kelley, N.D. "A Case for Including AtmosphericThermodynamic Variables in Wind Turbine LoadingParameters Identification," NREL/CP-500-26829,Golden, CO: National Renewable Energy Laboratory,1999.

3. Mallat, S. A Wavelet Tour of Signal Processing,Academic Press, San Diego, CA, 1998.

4. Tangler, J., Smith, B., Jager, D., McKenna, E. andAllread, J. "Atmospheric Performance Testing of theSpecial Purpose Thin Airfoil Family: PreliminaryResults," Proc. WindPower '89, NREL-/TP-257-3628, Golden, CO: National Renewable EnergyLaboratory,1989.

5. Kelley, N. D., Wright, A. D., and Osgood, R. M."AProgress Report on the Characterization andModeling of a Very Flexible Turbine Design," 18th

ASME/AIAA Wind Energy Symposium, NREL/CP-500-25513, Golden, CO: National Renewable EnergyLaboratory, October 1998.

6. Wright, A. D. and Kelley, N. D. "Validation of aModel for a Two-Bladed Flexible Wind TurbineSystem," Proc. 3rd ASME/JSME Joint FluidsEngineering Conference, San Francisco, CA, July 18-23, 1999.

7. Kelley, N. D. "Full Vector (3-D) Inflow Simulationin Natural and Wind Farm Environments Using anExpanded Version of the SNLWIND (Veers)Turbulence Code," NREL/TP-442-5225, Golden,CO: National Renewable Energy Laboratory, 1993.

8. Tangler, J., Smith, B., Kelley, N., and Jager, D."Measured and Predicted Rotor Performance for theSERI Advanced Wind Turbine Blades,” Proc.WindPower '91, NREL/TP-253-4673, Golden, CO:National Renewable Energy Laboratory, 1990.